2 research outputs found
Robust expansion and hamiltonicity
This thesis contains four results in extremal graph theory relating to the recent notion of robust expansion, and the classical notion of Hamiltonicity. In Chapter 2 we prove that every sufficiently large ‘robustly expanding’ digraph which is dense and regular has an approximate Hamilton decomposition. This provides a common generalisation of several previous results and in turn was a crucial tool in Kühn and Osthus’s proof that in fact these conditions guarantee a Hamilton decomposition, thereby proving a conjecture of Kelly from 1968 on regular tournaments.
In Chapters 3 and 4, we prove that every sufficiently large 3-connected -regular graph on vertices with ≥ n/4 contains a Hamilton cycle. This answers a problem of Bollobás and Häggkvist from the 1970s. Along the way, we prove a general result about the structure of dense regular graphs, and consider other applications of this.
Chapter 5 is devoted to a degree sequence analogue of the famous Pósa conjecture. Our main result is the following: if the largest degree in a sufficiently large graph on n vertices is at least a little larger than /3 + for ≤ /3, then contains the square of a Hamilton cycle